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e5c8b6cf-999c-4004-b899-becfeecb8bc4 | In the given diagram is a plot of scores of individual players from two teams, Jia and Yi, in a basketball game. The team that won the game is the ______ team. | Jia |
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3691469a-5f85-4231-ad01-c0c478726e09 | The diagram shows the top view and left view of a solid constructed with several small cubes. At most, how many small cubes could this solid possibly have? | $$9$$ |
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0752b238-0b62-4040-98d0-01523b793410 | As shown in the figure, in $$\triangle ABC$$, $$AB=AC$$, $$\angle BAC=54^{\circ}$$. The angle bisector of $$\angle BAC$$ and the perpendicular bisector of $$AB$$ intersect at point $$O$$. Point $$E$$ is on side $$BC$$, and point $$F$$ is on side $$AC$$. When $$\angle C$$ is folded over along $$EF$$, point $$C$$ coincides with point $$O$$. Find $$\angle OEC=$$___. | $$108^{\circ}$$ |
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f05ff959-25cb-44c9-b156-084ded4bf1d9 | As shown in the figure, in the trapezoid ABCD, AB ∥ DC, DE ∥ CB, the perimeter of △AED is 16cm, EB=3cm, then the perimeter of the trapezoid ABCD is ______. | 22cm. |
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7563b69f-9b41-4508-913f-d755452cba10 | If the partial graph of the quadratic function $$y=-x^{2}+2x+k$$ is as shown in the figure, then for the quadratic equation in one variable concerning $$x$$, $$-x^{2}+2x+k=0$$, one solution is $$x_{1}=3$$, and the other solution is $$x_{2}=$$___. | $$-1$$ |
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673e0436-f9ff-4656-81e7-ef475af70343 | As shown in the figure, in the cube $$ABCD-A_{1}B_{1}C_{1}D_{1}$$ with side length $$a$$, $$M$$ and $$N$$ are the midpoints of the edges $$A_{1}B_{1}$$ and $$B_{1}C_{1}$$, respectively. $$P$$ is a point on the edge $$AD$$ such that $$AP=\dfrac{a}{3}$$. The plane passing through $$P$$, $$M$$, and $$N$$ intersects the bottom face at $$PQ$$, where point $$Q$$ is on $$CD$$. Find the length of $$PQ=$$___. | $$\dfrac{2\sqrt{2}}{3}a$$ |
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76185125-3aa6-4499-aec0-620cd7dcdbd4 | The book "Algorithm General Principles" by Cheng Dawei is an important Chinese traditional mathematics work. In "Algorithm General Principles" it records: "When the Qiujian on the flat ground hasn't risen a thousand times, the pedal board is one chi above the ground. Sending two steps forward makes it even with the person, and it has been recorded that a person is five chi tall. Beautiful women compete to perch, in the end, laughing words bring joy. Skilled craftsmen with curiosity, can calculate how long the Qiujiansuo is?" $$1$$ step $$=5$$ chi. Translation: "When the Qiujian is stationary, the pedal board is $$1$$ chi high above the ground. If the pedal board of the Qiujian moves forward by two steps ($$10$$ chi), it becomes as tall as a person, and it is known that the person's height is $$5$$ chi. Beautiful maidens and talented people compete every day to perch the Qiujian, laughter and words continue throughout the day. Curious and skilled craftsmen can calculate how long the Qiujian's rope is?" As shown in the figure, assume that the rope length of the Qiujian always remains straight, $$OA$$ is the stationary state of the Qiujian, $$A$$ is the pedal board, $$CD$$ is the ground, point $$B$$ is the position of the pedal board after moving two steps, the arc $$AB$$ is the pedal board's moving track. It is known that $$AC=1$$ chi, $$CD=EB=10$$ chi, the person's height $$BD=5$$ chi. Set the rope length $$OA=OB=x$$ chi, then the equation can be formed as ___. | $$(x-4)^{2}+10^{2}=x^{2}$$ |
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b4cc1e66-f466-490f-8c9c-131e54d39a1c | As shown in the figure, ∠AOB = 45°, draw perpendiculars from points on OA at distances 1, 3, 5, 7, 9, 11 from point O to intersect with OB, obtaining a set of black trapezoids. Their areas are labeled as S$_{1}$, S$_{2}$, S$_{3}$, S$_{4}$, …, observing the pattern in the figure, find the area of the 10th black trapezoid S$_{10}$ = . | 76 |
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71b2ec81-5f9f-427a-a199-ffad95a997b0 | As shown in the figure, the area of the left view of a regular triangular prism with side length $2a$ is $\sqrt{3}{{a}^{2}}$. The lateral surface area of this regular triangular prism is . | $6{{a}^{2}}$ |
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78381632-94cc-4d47-b0bb-65ce07f36cd0 | As shown in the figure, AC is the diagonal of plane a, and AO = a, with angle AOC forming a 60° angle. OC = a, AA' is perpendicular to a at A', and angle A'OC = 45°. Find the distance from point A to the line OC. | $\frac{\sqrt{14}}{4}a$ |
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5c0c4d84-84b9-4c28-98e7-5a3f92da34d6 | As shown in the figure, consider the inverse proportional function $y=\frac{3}{x}$ with points $P_1(1, y_1), P_2(2, y_2), ..., P_n(n, y_n)$ on its graph. Draw vertical lines from these points to the x-axis, intersecting it at points $A_1, A_2, ..., A_n$. Connect $A_1P_2, A_2P_3, ..., A_{n-1}P_n$. Then, using $A_1P_1, A_1P_2$ as one pair of adjacent sides, draw a parallelogram $A_1P_1B_1P_2$. Similarly, using $A_2P_2, A_2P_3$ as one pair of adjacent sides, draw a parallelogram $A_2P_2B_2P_3$, and so on. Find the vertical coordinate of point $B_n$. (Express the result with a formula containing $n$) | $\frac{6n+3}{n\left( n+1 \right)}$ |
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3c031dff-e91b-4d2e-90fd-70c0ea0714da | As shown in the figure, in right triangle ABC, ∠C=90°, ∠ABC=30°, point D is on side BC, CD=1. Triangle ACD is folded along the line AD such that point C falls on point E on side AB. If point P is a moving point on the line AD, what is the minimum value of PB + PE? | 3. |
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f7e2e04e-1046-4505-b753-064cde8406b7 | As shown in the diagram, point A represents the intersection of Street 2 and Avenue 4, while point B represents the intersection of Street 4 and Avenue 2. If the path from A to B is represented by (2,4)→(2,3)→(3,3)→(4,3)→(4,2), please represent another path from A to B using the same method: ______. (Write down another path) | (2,4)→(3,4)→(4,4)→(4,3)→(4,2). |
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9a490ac7-e398-4ca2-831d-799712c8a7e0 | As shown in Figure 1, a cube with an edge length of 1 is cut along the diagonal plane (the shaded part in the figure) into two pieces and reassembled into the oblique quadrangular prism shown in Figure 2. The surface area of the resulting oblique quadrangular prism is
Figure 1 Figure 2 | $4+2\sqrt{2}$ |
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2efd81ce-a0cb-4113-964e-0dc63209ef64 | As shown in the figure, it is the net of a cube, and the face marked with the letter $$A$$ is the front face of the cube. If the values marked on opposite faces of the cube are additive inverses, find the value marked by the letter $$A$$. | $$-3$$ |
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698173f5-9ef7-49a3-803d-2afb4d302eae | As shown in the figure, there is a rectangular flower bed $$ABCD$$ with uneven edges. One side, $$AD$$, uses an existing wall, and the total length of the bamboo surrounding the other three sides is $$6m$$. If the area of the rectangle is $$4m^{2}$$, then the length of $$AB$$ is ___$$m$$. | $$1$$ |
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c39e04ad-2abe-4d23-8521-a5c643f47be7 | As shown in the figure, in trapezoid $ABCD$, $AB\parallel CD, AB=4, AD=3, CD=2$, $\overrightarrow{AB}\cdot \overrightarrow{AD}=\frac{3}{2}$, $\overrightarrow{AM}=\lambda \overrightarrow{AD}$, $\left( \lambda \in \left( 0, 1 \right) \right)$, and $\overrightarrow{AC}\cdot \overrightarrow{BM}=-3$, find the value of $\lambda$. | $\frac{2}{3}$ |
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62dceff3-83c7-45d7-ba20-d2859f7b2c6e | Look at the picture and complete the equation: ______$$+$$______$$=$$______. ______$$+$$______$$=$$______. ______$$-$$______$$=$$______. ______$$-$$______$$=$$______. | 4
6
10
6
4
10
10
4
6
10
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4 |
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cfa93b40-c380-44f4-a318-cad069687b35 | A cube labeled with the numbers $1$, $1$, $2$, $3$, $3$, $5$ is unfolded into a net as shown in the figure. After rolling the cube once, let the number facing up be $x$, and the number facing down be $y$. This results in a point $\left( x,\,y \right)$ in the rectangular coordinate system of the plane. It is known that a line determined by two points obtained from Xiaohua's first two rolls passes through the point $P\left( 0,\,-1 \right)$. What is the probability that the point obtained from his third roll also lies on this line? | $\frac{2}{3}$ |
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69cb7234-c321-4031-b16b-198f9df1216f | As shown in the figure, this is the net of a cube. Which number is on the face opposite to the face numbered $2$? | $5$ |
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eb9fba46-ec54-4f71-ab9b-b23b770005b1 | In order to understand the water usage situation of the community residents, Xiao Jiang conducted a survey of 80 households. He found that the frequency of per capita daily water usage within the basic standard (50 liters) is 75%. So, the number of households exceeding the standard is ______. | The frequency of per capita daily water usage within the basic standard (50 liters) is 75%, hence the frequency of exceeding the standard is 1-75%. Therefore, the number of households exceeding the standard is: 80× (1-75%)=20. |
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23732edd-2276-43df-972f-8295e9989054 | As shown in the figure, in a rectangular coordinate system, the acute angle formed by line l and the positive half-axis of the y-axis is 60°. Passing through point A(0,1), line l intersects the vertical line of the y-axis at point B. Through point B, the vertical line of line l intersects the y-axis at point A$_{1}$. Taking A$_{1}$B and BA as adjacent sides, construct the parallelogram ABA$_{1}$C$_{1}$; through point A$_{1}$, the vertical line of the y-axis intersects line l at point B$_{1}$. Through point B$_{1}$, the vertical line of line l intersects the y-axis at point A$_{2}$. Taking A$_{2}$B$_{1}$ and B$_{1}$A$_{1}$ as adjacent sides, construct the parallelogram A$_{1}$B$_{1}$A$_{2}$C$_{2}$; ...; continuing this method, the coordinates of C$_{3}$ are: and the coordinates of C$_{n}$ are | $(-16\sqrt{3},64)$
$(-\sqrt{3}\times {{4}^{n-1}},{{4}^{n}})$ |
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24335bff-7ac8-44b5-a3e2-847cf67fec56 | As shown in the figure, in the quadrilateral $$ABCD$$, $$AC=BD=6$$, and $$E$$, $$F$$, $$G$$, $$H$$ are the midpoints of $$AB$$, $$BC$$, $$CD$$, and $$DA$$ respectively. Connect $$EG$$ and $$FH$$. Suppose $$EG$$ and $$FH$$ intersect at point $$O$$, then the value of $$EG^{2}+FH^{2}$$ is ___. | $$36$$ |
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d1621cec-cf9f-4e08-b70e-43cb166e6390 | As shown in the figure, in $\vartriangle ABC$, $\angle A={{64}^{\circ }}$, the angle bisectors of $\angle ABC$ and $\angle ACD$ intersect at point ${{A}_{1}}$, find $\angle {{A}_{1}}$; the angle bisectors of $\angle {{A}_{1}}BC$ and $\angle {{A}_{1}}CD$ intersect at point ${{A}_{2}}$, find $\angle {{A}_{2}}$; ...the angle bisectors of $\angle {{A}_{2}}BC$ and $\angle {{A}_{2}}CD$ intersect at point ${{A}_{3}}$, then $\angle {{A}_{3}}=$ . | ${{8}^{\circ }}$ |
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76106c56-4bfd-4acc-99ca-c9cc24e899c7 | As shown in the diagram, $$l$$ is the axis of symmetry of the quadrilateral $$ABCD$$. If $$AD \parallel BC$$, you have the following conclusions: 1. $$AB \parallel CD$$; 2. $$AB = BC$$; 3. $$AB \perp BC$$; 4. $$AO = OC$$. Among them, the correct conclusions are ______. (Fill in the numbers of the conclusions that you believe are correct) | 1.2.4. |
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4699d550-7db6-417e-b961-a4b0d0e8cd77 | As shown in the figure, it is known that $$AB$$ is the diameter of circle $$\odot O$$, points $$C$$ and $$D$$ are on $$\odot O$$, and $$\angle ABC=35^{\circ}$$. Find $$\angle D=$$___. | $$55^{\circ}$$ |
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bb69a2ec-62ae-45cd-8986-2de943a3780b | As shown in the figure, in the regular dodecagon $$A_{1}A_{2}\cdots A_{12}$$, connect $$A_{3}A_{7}$$ and $$A_{7}A_{10}$$, then $$\angle A_{3}A_{7}A_{10}=$$___$$^{\circ}$$. | $$75$$ |
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9b4cda47-5994-4b29-8dde-aa832b1ec2a1 | If the fractional equation has an extraneous root, then the value of a is ______. | Multiply both sides of the equation by (x-4), yielding x=2(x-4)+a. Because the original equation has an extraneous root, x=4 is the simplest solution, and when x=4, a=4. Therefore, a is 4. |
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ee50cf8a-489a-4732-8125-a2275ca5e95e | As shown in the figure, in $\Delta ABC$, $D$ and $E$ are the midpoints of $AB$ and $AC$, respectively. Point $F$ is on $DE$, and $AF\bot CF$. If $AC=3$ and $BC=5$, what is $DF$? | 1 |
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6203de0f-e4cc-4910-908c-3a663eb3241e | As shown in the figure, among the common geometric bodies such as cones, cylinders, spheres, and rectangular prisms, which geometric bodies have identical main and left side views? (Fill in the number) | 1.2.3. |
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3ec26bd3-89c0-4465-947e-cb9777ac2acd | As shown in the figure, in triangle ABC, BD and CE are the two angle bisectors of triangle ABC. If ∠A = 52°, then the degree of ∠1 + ∠2 is | 64° |
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69494865-505a-4ad1-8fa2-53809e5bad0b | As shown in the figure, in the quadrangular pyramid $$P-ABCD$$, $$ABCD$$ is a parallelogram, and $$AC$$ and $$BD$$ intersect at $$O$$. $$G$$ is a point on $$BD$$ such that $$BG=2GD$$. Given $$\overrightarrow{PA}=\boldsymbol{a}$$, $$\overrightarrow{PB}=\boldsymbol{b}$$, $$\overrightarrow{PC}=\boldsymbol{c}$$, express the vector $$\overrightarrow{PG}=$$___. | $$\dfrac{2}{3}\boldsymbol{a}-\dfrac{1}{3}\boldsymbol{b}+\dfrac{2}{3}\boldsymbol{c}$$ |
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ff91583f-6ecc-46cc-ab18-23648c71193b | As shown in the figure, the triangle $OAB$ is a certain plane figure drawn using the oblique projection method to create an orthographic projection. What is the original area of the plane figure? | $4$ |
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51842346-9143-441b-9f3b-8193fe4c2a9a | As shown in the figure, line AB intersects with CD at point O, and OE is perpendicular to AB. If ∠AOD = 65°, then ∠EOC = ______. | ∵ ∠AOD = ∠COB (vertically opposite angles are equal), ∠AOD = 65°, ∴ ∠COB = 65°;
∵ OE ⊥ AB, ∴ ∠EOC = 90° - ∠COB = 25°.
Therefore, it is: 25°. |
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8fe11865-c927-46ff-8649-25fccaa9140b | As shown in the figure, the equation of the tangent line at point P of the function is, therefore,. | 1 |
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5202369d-1417-4c8e-808c-882436e71f53 | As shown in the figure, two identical right-angled triangular boards with a $$30^{\circ}$$ angle are stacked together, and $$\angle DAB=30^{\circ}$$. There are four conclusions: 1. $$AF\bot BC$$; 2. $$\triangle ADG\cong \triangle ACF$$; 3. $$OB=OE$$. The correct sequence of conclusions is ___. | 1.2.3. |
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c9ae5be7-c2ab-43e0-8933-f13243f0d5dc | As shown in the figure, in the plane rectangular coordinate system, given point A(1, 1), B(-1, 1), C(-1, -2), D(1, -2), one end of a fine line with a length of 2015 units and no elasticity (ignoring the thickness of the line) is fixed at point A. The line is wound tightly along the edges of quadrilateral ABCD in the order of A→B→C→D→A... Where is the coordinate of the other end of the line located? ______ | (-1, -2). |
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b308b043-19c8-4051-8e2f-38c30cc72e6e | As shown in the figure, AD is the median of triangle ABC, and point E is the midpoint of AD. If the area of triangle AEC is 12 cm$^{2}$, then the area of triangle ABC is ______ cm$^{2}$. | 48. |
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8d9f230a-e174-40ba-8b1b-92900db2addc | As shown in the figure, in the square ABCD, AB=4, E is the midpoint of BC. Two moving points M and N are on sides CD and AD, respectively, and MN=1. If triangle ABE is similar to the triangle with vertices D, M, and N, then DM is. | $\frac{\sqrt{5}}{5}$ or $\frac{2\sqrt{5}}{5}$ |
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e55844f0-f544-4720-bc5f-aa092b1430ed | In the diagram below, there are ______ parallelograms and ______ triangles. | 6
6 |
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c1f299ae-4012-419e-9194-691b7869ee08 | As shown in the figure, in the cube $$ABCD-A_{1}B_{1}C_{1}D_{1}$$, $$E$$, $$F$$, $$G$$, $$H$$ are the midpoints of segments $$CC_{1}, C_{1}D_{1}, D_{1}D, CD$$ respectively. $$N$$ is the midpoint of $$BC$$, and point $$M$$ moves within the quadrilateral $$EFGH$$ and its interior. When $$M$$ satisfies ___, there is $$MN\parallel$$ plane $$B_{1}BDD_{1}$$. | $$M\in FH$$ |
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842d313e-2ea9-48b9-a16f-b6ebead655b4 | The line $y=x$ intercepts a chord on the circle ${{x}^{2}}+{{\left( y-2 \right)}^{2}}=4$ with a length of | $2\sqrt{2}$ |
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614b1de6-17eb-4325-8dbb-712015e7dc7d | Given the parametric equation of curve C$_{1}$ (φ is a parameter), with the origin as the pole and the positive x-axis as the polar axis to establish the polar coordinate system, the polar coordinate equation of curve C$_{2}$ is ρ=2. The vertices of square ABCD all lie on C$_{2}$, and A, B, C, and D are arranged in counterclockwise order. The polar coordinate of point A is (2,), let P be any point on C$_{1}$, then the range of values of |PA|$^{2}$+|PB|$^{2}$+|PC|$^{2}$+|PD|$^{2}$ is ______. | [32,52]. |
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4bd9b564-c85c-4290-a59d-a2f148f15068 | Given the hyperbola $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$ with $a > 0, b > 0$, let ${{A}_{1}},{{A}_{2}}$ be the left and right vertices, $F$ be the right focus, and $B$ be the top endpoint of the imaginary axis. If there exist two distinct points ${{P}_{i}}(i=1,2)$ on the line segment $BF$ (excluding the endpoints) such that $\overrightarrow{{{P}_{i}}{{A}_{1}}}\cdot \overrightarrow{{{P}_{i}}{{A}_{2}}}=0$, then the range of the eccentricity of the hyperbola is . | $\left( \sqrt{2},\frac{\sqrt{5}+1}{2} \right)$ |
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2a0d65bf-e803-4120-9397-a4401a347e3b | As shown in the figure, point D is a point on the side BC of $\vartriangle ABC$, $\overrightarrow{BD}=2\overrightarrow{DC}$, ${{E}_{n}}\left( n\in {{N}^{*}} \right)$ are points on AC, satisfying $\overrightarrow{{{E}_{n}}A}=\left( 3{{a}_{n}}-3 \right)\overrightarrow{{{E}_{n}}D}+\left( -{{n}^{2}}-n+1 \right)\overrightarrow{{{E}_{n}}B}$, where the real number sequence $\left\{ {{a}_{n}} \right\}$ satisfies ${{a}_{1}}=2$, then $\frac{1}{{{a}_{1}}}+\frac{1}{{{a}_{2}}}+\frac{1}{{{a}_{3}}}+\cdots +\frac{1}{{{a}_{n}}}=$. | $\frac{n}{n+1}$ |
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64c76a22-d978-4db8-9666-7a6989a5df38 | In a math exam, the number of students and their average scores in Grade 9 (Class 1) and (Class 2) are shown in the table below. What is the average score of these two classes combined? | $$82.6$$ |
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797da825-0224-4a33-9504-7d7df2f30ba4 | As shown in the figure, it is known that angles α and β share the vertex O, and α + β < 180°, α = $\frac{1}{3}$β. If the supplementary angle of β is equal to $\frac{3}{2}$α, then β = ? degrees. | 120 |
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6cfd53ae-a67e-4b81-9956-bbbd71908ad5 | Given that the positions of real numbers a and b on the number line are as shown in the figure, determine $\frac{\text{a-b}}{\text{a}+\text{b}}$ (fill in “>”, “=” or “<”) | > |
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ba291fc5-4fd0-4157-8bd4-17ab68e74079 | In January 2013, Beijing experienced the month with the most foggy days in 59 years. According to meteorological statistics, from January 1st to January 30th, 2013, there were 26 days with foggy weather in Beijing. The 'Ambient Air Quality Index (AQI) Technical Regulation (Trial)' categorizes the air quality index into six levels, as shown in Table 1. Table 1: [Image](fa455f3c3a46e6321e41aa358b1398c5217d5acceaf7485df2500279661efc54) Table 2 shows the recorded AQI and the visibility level y (in km) at an observational meteorological station over four consecutive days. Table 2: [Image](4a3cd719ced9a0e0ff96979c57b43a1e6dc77b9b3126dc2bf3c34b8d57d6a405). Let x = M/100, where M is the AQI. Based on the data from Table 2, the regression equation of y in terms of x is ___. | $$\hat {y}=10.25-1.05x$$ |
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4b9b4d43-1885-438f-9ede-358b6fe93b0b | As shown in the figure, it is known that $$\angle AOB=120^{\circ}$$. A line is drawn through a point $$M$$ on the angle bisector, intersecting rays $$OA$$ and $$OB$$ at points $$P$$ and $$Q$$, respectively. If $$|OM|=1$$, then the minimum value of $$\left \lvert OP\right \rvert +2\left \lvert OQ\right \rvert $$ is ___. | $$3+2\sqrt{2}$$ |
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36b1cf83-dea1-42fe-94bd-446029d2d389 | As shown in the figure, it is known that the coordinates of the three vertices of $$\triangle ABC$$ are $$A(0,4)$$, $$B(-1,1)$$, $$C(-2,2)$$ respectively. Translate $$\triangle ABC$$ to the right 4 units to obtain $$\triangle A'B'C'$$. The corresponding points of points $$A$$, $$B$$, $$C$$ are $$A'$$, $$B'$$, $$C'$$ respectively. Then rotate $$\triangle A'B'C'$$ counterclockwise by $$90^{ \circ }$$ around point $$B'$$ to obtain $$\triangle A''B''C''$$. The corresponding points of points $$A'$$, $$B'$$, $$C'$$ are $$A''$$, $$B''$$, $$C''$$ respectively. The coordinates of point $$A''$$ are ___. | $$(6,0)$$ |
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1d5096e3-e96e-4564-a015-d69210731b7a | As shown in the figure, the relationship of the masses of the three objects a, b, and c is ______. | a > b > c. |
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583f242c-65d0-45f1-ad55-9026cfad9c46 | As shown in the figure, in $$ \triangle ABC$$, $$ \angle ABC=90^{ \circ }$$, $$AB=3$$, $$BC=4$$, point $$O$$ is the midpoint of $$BC$$. A semicircle with diameter $$BC$$ intersects $$AC$$ and $$AO$$ at points $$M$$ and $$N$$, respectively. Find $$AN=$$___, $$\dfrac{AM}{MC}=$$___. | $$\sqrt{13}-2$$ $$\dfrac{9}{16}$$ |
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73ffc315-dce3-4145-ba17-9bc77752943e | As shown in the figure, fold rectangle $$ABCD$$ along $$EF$$ so that vertex $$C$$ falls exactly on the midpoint $$C'$$ of side $$AB$$, and point $$D$$ falls at $$D'$$. $$C'D'$$ intersects $$AE$$ at point $$M$$. If $$AB=6$$, $$BC=9$$, then the length of $$AM$$ is ___. | $$\dfrac94$$ |
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53a4bd69-dd0e-4be5-ac1e-49c93e250b9f | In a singing competition held at a certain school, the scores given by seven judges to a student are shown in the stem-and-leaf plot. After removing the highest and lowest scores, the variance of the remaining data is shown. | 2 |
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f70d3f5c-16d6-4b97-9ca6-f4b3fadee8cc | As shown in the figure, for an arbitrary quadrilateral $$ABCD$$, the midpoints of each side are $$E$$, $$F$$, $$G$$, and $$H$$, respectively. If the diagonals $$AC$$ and $$BD$$ are both $$20cm$$ long, then the perimeter of quadrilateral $$EFGH$$ is ___ $$cm$$. | $$40cm$$ |
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0c912375-5669-49c8-91b1-8e7ad37ca202 | As shown in the figure, in triangle ABC, CD bisects ∠ACB and intersects AB at D, DE∥BC intersects AC at point E, AC=6, DE=4, find BC. | 12. |
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3462e5c9-9bfa-41d9-9f12-3a7682420a88 | As shown in the figure, if the position of point A is (-1, 0), then the positions of points B, C, D, and E are respectively , , , | (-2, 3)
(0, 2)
(2, 1)
(-2, 1) |
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dd3da7d4-1abb-4ebc-bba2-def30cbe848d | As shown in the flowchart, if the input value of $$x$$ is $$1$$, then the output result is ______. | $$4$$ |
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db86987c-d5e2-4950-9bf5-1d2d486bb4ed | As shown in the figure, the radius of circle $$ \odot O$$ is $$2$$, $$OA=4$$, $$AB$$ intersects $$ \odot O$$ at point $$B$$, chord $$BC \parallel OA$$, connect $$AC$$. Find the area of the shaded region in the figure. | $$\dfrac{2}{3} \pi $$ |
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03e8c240-29bb-477b-bea5-6ab9f43182cd | As shown in the figure, in a rectangular coordinate system on a plane, $$\triangle A_{1}A_{2}A_{3}$$, $$\triangle A_{3}A_{4}A_{5}$$, $$\triangle A_{5}A_{6}A_{7}$$, $$\triangle A_{7}A_{8}A_{9}$$, $$\cdots$$, are all equilateral triangles. Moreover, the coordinates of the points $$A_{1}$$, $$A_{3}$$, $$A_{5}$$, $$A_{7}$$, $$A_{9}$$ are respectively $$A_{1}(3,0)$$, $$A_{3}(1,0)$$, $$A_{5}(4,0)$$, $$A_{7}(0,0)$$, $$A_{9}(5,0)$$. According to the pattern reflected in the figure, the coordinates of $$A_{100}$$ are ___. | $$\left (\dfrac{5}{2},-\dfrac{51\sqrt{3}}{2} \right )$$ |
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a462dd9d-7e33-43e8-881c-ac80aad5ef19 | As shown in the figure, in the rhombus ABCD, ∠ABC = 60°, point E is the midpoint of CD, BE intersects AC at point F. If AB = 4, then the length of AF is . | $\frac{8}{3}$ |
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c3c2563a-231d-4bee-9c51-2e5a6aa521c2 | The figure shows a sign placed at the entrance of a senior activity center. This sign is formed by three large dice piled together. Each die has six faces with points ranging from 1 to 6. Among them, 7 faces are visible, and the remaining 11 faces are not visible. What is the total sum of the points on the non-visible faces? | 39 |
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da852a66-7be8-46dd-acfb-e4cb2e5e1007 | Xiao Jing draws shapes using circles, triangles, and squares following a certain pattern. The first eight shapes are shown in the figure. Predict how many squares the $$\number{2017}$$th shape contains. | $$336$$ |
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a7e1af2c-2aeb-4a6b-9fd2-9c173f58a066 | Observe the following equation: In the above numerical pyramid, counting from top to bottom, the number $$\number{2016}$$ is in layer ___. | $$44$$ |
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537bf168-6626-499e-bdde-8925ba2f649a | In response to the call for 'scholarly campus' construction, to cultivate a good cultural reading atmosphere throughout the school, students randomly surveyed the average weekly reading time of 40 students. The statistical results are shown in the figure. According to the information, the median reading time in this survey is ___ hours, and the mode is ___ hours. | 6 6 |
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9812c4d3-879f-44a1-8b3c-27595fc8c017 | The number represented by point A on the line is ( ), the decimal represented by point B is ( ), and the fraction represented by point C is ( ). | −2
0.5
$1\frac{1}{4}$ |
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ac690b17-6e8b-4d5a-83a4-6e88cfcbadcd | During the 'National Reading Month' event, Xiaoming surveyed the expenditure plans for purchasing extracurricular books of $$40$$ classmates in the class this semester, and compiled the results into the statistical chart as shown. Please answer the following questions based on the relevant information (directly fill in the results). [](0dc17eafe21053736f4fbd2978f3058766dcece5f75ca5701a5733e9acac8b0b)(1) The mode of the sampled data obtained in this survey is ___; (2) The median of the sampled data obtained in this survey is ___; (3) If there are $$1000$$ students in the school, estimate the number of students planning to spend $$50$$ yuan on purchasing extracurricular books this semester based on the sample data, and fill in ___. | (1)$$30$$ yuan (2)$$50$$ yuan (3)$$250$$ |
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2ccf9aff-8c42-49bd-9bfb-00e0cfb701f4 | The average daily temperature of sites A and B in early September is shown in the figure. Determine the relationship between the variance of the average daily temperature over these 10 days at sites A and B: $s_{ ext{A}}^{2}$$___$$s_{ ext{B}}^{2}$ (fill in ">" or "<"). | > |
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fa944e01-6ec2-44e7-9bab-10b403154cb4 | As shown in the figure, AB∥CD∥EF, AD:DF=3:2, BC=6, then the length of CE is . | 4 |
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c7693f6d-9c02-4418-9cdf-8096a0f7ac8c | As shown in the figure, in $$\triangle ABC$$, $$DE\parallel BC$$, and $$AD=2$$, $$DB=3$$, find $$\dfrac{DE}{BC}=$$___. | $$\dfrac{2}{5}$$ |
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dd3e4b71-6f5e-416f-98ea-8c83a518b307 | As shown in the figure, part of the trajectory of a thrown object is represented by the curve $$y_{1}=ax^{2}+bx+c$$ with $$a \neq 0$$. The vertex of the trajectory is at point $$A(1,3)$$ and it intersects the x-axis at point $$B(4,0)$$. A line $$y_{2}=mx+n$$ (where $$m\neq 0$$) also passes through points $$A$$ and $$B$$. The following conclusions can be made: 1) $$2a+b=0$$; 2) $$abc>0$$; 3) $$b^{2}-4ac>0$$; 4) The other intersection of the trajectory with the x-axis is $$(-1,0)$$; 5) When $$1<x<4$$, $$y_{2}<y_{1}$$; 6) The equation $$ax^{2}+bx+c=3$$ has two equal real roots. Among these, the correct ones are ______. | 1.3.5.6. |
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f8ee9b15-8735-4eaf-84c0-810a61f4748d | As shown in the figure, given that $$OB=1$$, construct an isosceles right triangle $$A_{1}BO$$ with $$OB$$ as one leg, then construct an isosceles right triangle $$A_{2}A_{1}O$$ with $$OA_{1}$$ as one leg, and so on. The length of the line segment $$OA_{n}$$ is ___. | $$\sqrt{2^{n}}$$ |
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92786873-0462-46d3-83fe-8e9c6b77e417 | As shown in the figure, in the right triangle $$Rt\triangle ABC$$, $$\angle B=90^{\circ}$$, $$AB=4$$, $$BC>AB$$, point $$D$$ is on $$BC$$, and in the parallelogram $$ADCE$$ where $$AC$$ is the angle bisector, the minimum value of $$DE$$ is ___. | $$4$$ |
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82a49a3e-486a-4d1e-9211-e77a454f07ae | Given a quadratic function $$y=ax^{2}+bx+c$$($$a\not =0$$, $$a$$, $$b$$, $$c$$ are constants), its axis of symmetry is the line $$x=1$$. The corresponding values of the independent variable and the function value $$y$$ are shown in the table below. Write down an approximate value of a positive root of the equation $$ax^{2}+bx+c=0$$ (to the precision of $$0.1$$). | $$2.2$$ (not unique, any value close to it is acceptable) |
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f5485c75-88d2-4652-b453-8aa2272893cb | Look at the diagram and fill in the blanks.
1. AB is () of AE.
2. AC is () of AE.
3. BC is () of BE. | $\frac{1}{4}$
$\frac{1}{2}$
$\frac{1}{3}$ |
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1701e4f1-9ef7-4077-a147-2dd827f80591 | As shown in the figure, in the rectangular coordinate system on the plane, a point $P(1,1)$ on the line $y=x$, a point $C$ on the y-axis, connect $PC$, the line segment $PC$ rotates 90° clockwise around the point $P$ to the line segment $PD$, if a vertical line $AB$ is drawn through point $D$ perpendicular to the x-axis, with $B$ as the foot, and line $AB$ intersects the line $y=x$ at point $A$, and $BD=2AD$, connect $CD$, if the line $CD$ intersects the line $y=x$ at point $Q$, the coordinates of point $Q$ are ___. | $$ \left ( \dfrac{9}{4} , \dfrac{9}{4} \right ) $$ |
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e95fb5ff-27a0-493a-8a80-f6d79f57087a | The square ABCD with side length $\sqrt{5}$ and a right-angled triangle board is placed as shown in the figure. Extend CB to intersect with one of the right-angled sides of the triangle board at point E. Then the area of quadrilateral AECF is. | 5 |
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8d8befaf-7998-499e-a6ce-60397ae72829 | As shown in the figure, in the isosceles right triangle $$\triangle ABC$$, $$\angle C = 90^{\circ}$$, and $$AC = 12$$. Point $$D$$ is a point on $$AC$$. If $$\tan \angle DBA = \dfrac{1}{5}$$, then the length of $$AD$$ is ___. | $$4$$ |
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bb047095-da86-4480-b48d-ac98f3a64818 | As shown in the figure, the largest triangle is an equilateral triangle with side length 2. The midpoints of each side of this triangle are connected to form the second triangle. Continuing in this manner, a total of 10 triangles are formed. The sum of the areas of these 10 triangles is what? | $\frac{4\sqrt{3}}{3}\left( 1-\frac{1}{{{4}^{10}}} \right)$ |
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07cb2347-63f1-4ebc-9b45-50319382afae | As shown in the figure, the first shape is an equilateral triangle ABC made by joining 3 wooden sticks of the same length, and the second shape is made by joining 5 identical sticks; according to the pattern shown in the figure, in the nth shape, the number of such wooden sticks required is ______. | 2n+1. |
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ba1b6698-e282-4820-a95b-a913d45cc018 | As shown in the figure, if $AB \parallel CD \parallel EF$, then the equation among \( \angle \alpha \), \( \angle \beta \), and \( \angle \gamma \) is. | \( \angle \beta = \angle \alpha + \angle \gamma \); |
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55cbde2a-0949-4077-9941-e3d3ddb431a9 | Based on past sales records of a certain type of bread, a bakery created a histogram of daily sales frequency (as shown in the figure). Assuming a month is calculated as 30 days, estimate the number of days in a month that the bakery sells at least 150 pieces of bread as ___. | 9 |
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1f4bb35b-99a2-4b0e-a060-9ea9bbbe183e | Given the equation of circle $$\odot O$$ as $$x^{2}+y^{2}-2=0$$, and the equation of circle $$\odot O'$$ as $$x^{2}+y^{2}-8x+10=0$$, as shown in the figure. From the moving point $$P$$, draw tangents $$PA$$ and $$PB$$ to the circles $$\odot O$$ and $$\odot O'$$, respectively. The points of tangency are $$A$$ and $$B$$, and $$PA=PB$$. Then the trajectory equation of the moving point $$P$$ is ___. | $$x=\dfrac{3}{2}$$ |
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2f52889c-9b6f-4925-8245-ed90d2d5a64f | As shown in the figure, $AB$ is a diameter of circle $\odot O$, $C$ and $D$ are two points on $\odot O$, and $C$ is the midpoint of arc $\overset\frown{AD}$. Given that $\angle BAD = 20{}^\circ$, what is the measure of $\angle ACO$? | 55° |
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0b02c031-bdbd-4bd8-9de2-fe81c98a42ad | As shown in the figure, a park plans to use pebbles to pave two intersecting 'health stone paths' (line segments $AD$ and $CE$), and build 'flower gallery corridors' (line segments $AC$ and $ED$) between the two ends of these 'health stone paths' for citizens to relax and exercise. The completed section $BD=20m$, $ED=10\sqrt{6}m$, $\angle BED={{45}^{\circ }}$ (\Delta BDE$ is an acute triangle). Due to design requirements, the total length of the unfinished sections $AB$ and $BC$ can only be $40m$. Find the shortest length of the remaining 'flower gallery corridor' (line segment $AC$) as $m$. | 20 |
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c27bcb88-6870-4b55-bf41-dec2f8e108b1 | As shown in the figure, in $$\triangle ABC$$, $$D$$ and $$E$$ are points on $$AB$$ and $$AC$$, respectively. Point $$F$$ is on the extension line of $$BC$$. $$DE \parallel BC$$, $$\angle A=46^{ \circ }$$, $$\angle 1=52^{ \circ }$$, then $$\angle 2=$$___ degrees. | $$98$$ |
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cdf63008-bc36-49f7-ac46-7093eda8fb3c | As shown in the figure, it is known that $$Rt\triangle A'BC'\cong Rt\triangle ABC$$, $$\angle A'C'B=\angle ACB=90^{\circ}$$, $$\angle A'BC'=\angle ABC=60^{\circ}$$. The right triangle $$Rt\triangle A'BC'$$ can be rotated around point $$B$$. When the three points $$A$$, $$C'$$, $$A'$$ are collinear, $$\angle ABC'=$$___. | $$60^{\circ}$$ |
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21b98b0d-c8b6-477d-829b-749e7e165994 | To prevent 'hand-foot-mouth disease', a certain school conducts 'fumigation disinfection' in classrooms. During the disinfection period, the function relationship between the drug concentration $$y(\text{mg})$$ in the air per cubic meter and time $$x$$ (minutes) is depicted in the figure. It is known that during the drug combustion phase, $$y$$ is directly proportional to $$x$$, and after combustion, $$y$$ is inversely proportional to $$x$$. It is measured that the combustion completes in $$10$$ minutes, during which the drug concentration in the classroom's air per cubic meter is $$8\ \text{mg}$$. The drug concentration must be below $$1.6\ \text{mg}$$ per cubic meter for it to be non-toxic to humans. So, starting from the disinfection, after ___ minutes, will the air in the classroom meet safety requirements. | $$50$$ |
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1de87aa0-bb30-4c27-bfec-218aec84ef4b | As shown in the figure, the line y = x + 1 intersects the x-axis and y-axis at points A and B, respectively. Taking point A as the center of the circle, draw an arc with the length of AB as the radius to intersect the x-axis at point A$_{1}$. Then, draw a vertical line through point A$_{1}$ on the x-axis to intersect the line at point B$_{1}$. Taking point A as the center of the circle, draw an arc with the length of AB$_{1}$ as the radius to intersect the x-axis at point A$_{2}$... Continuing this method, the coordinates of point A$_{8}$ are . | (15,0) |
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18f1fbf0-5515-4fbd-8b5d-2fc30346e7dc | Little fox departs from home, first walking ______ meters towards the ______ direction to reach the Juvenile Palace, then walking ______ meters towards the ______ direction to arrive at Huashan Park. | north
40
east
50 |
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006d89e1-9bfc-43c8-a2d2-581ba5ede4cd | As shown in the figure, $\Delta ABC$ is an isosceles right triangle, and $BC$ is the hypotenuse. Point $P$ is a point inside $\Delta ABC$, $AP=3$. Connect $PB$, and rotate $\Delta ABP$ to the position of $\Delta ACQ$, then the length of $PQ$ is. | $3\sqrt{2}$ |
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b01fa7d8-b2cc-426e-880a-bc81a72dc7e4 | As shown in the figure, in triangle ABC, AB=20, AC=12, BC=16. The triangle ABC is folded so that AB falls onto the straight line AC. What is the area of the overlapping part (shaded area)? | 36 |
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f8ee5f17-d3df-4a79-aad4-91158cf86beb | As shown in the figure, the shaded part represents the set ______. | (A∩B)∪(C∪(A∪B)). |
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55fa79f3-ccf3-429c-9a2f-27601bee8d22 | As shown in the figure, D is the midpoint of BC. Compare the area of △ABD and △ACD. | Equal |
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0387084e-6dac-45c8-992c-a64d2717755d | As shown in the figure, if $$\angle 1 + \angle 2 = 222^{\circ}$$, then $$\angle 3=$$___. | $$69^{\circ}$$ |
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c0dc5802-3cfd-4330-ba68-e773e589c0ae | As shown in the figure, in the parallelogram $$ABCD$$, point $$E$$ is on $$AB$$, and $$CE$$ and $$BD$$ intersect at point $$F$$. If $$AE:BE=4:3$$ and $$BF=2$$, then $$DF=$$___. | $$\dfrac{14}{3}$$ |
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865a31e6-b901-4aaa-9225-b09256c5bcd8 | Observe the following three inequalities: 1.; 2.$\left( {{7}^{2}}+{{9}^{2}}+{{10}^{2}} \right)\left( {{6}^{2}}+{{8}^{2}}+{{11}^{2}} \right)\ge {{\left( 7\times 6+9\times 8+10\times 11 \right)}^{2}}$; 3. If , then the minimum value of is. | $\frac{\text{2}}{\text{3}}$ |
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39a58245-5ed3-46c9-8d23-4ec1327ef66d | As shown in the figure, in $$\triangle ABC$$, $$AB=12$$, $$AC=5$$, $$AD$$ is the angle bisector of $$\angle BAC$$, $$AE$$ is the median on side $$BC$$, draw $$CF \perp AD$$ at point $$F$$ through point $$C$$, connect $$EF$$, then the length of segment $$EF$$ is ___. | $$3.5$$ |
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7eb3f510-035e-46b4-9f33-061b498050f4 | As shown in the figure, if the point $$M$$ on the ellipse $$\dfrac{x^{2}}{9}+\dfrac{y^{2}}{25}=1$$ has a distance of $$3$$ to the focus $$F_{1}$$, then the perimeter of $$\triangle MOF_{1}$$ is ___. | $$7+\sqrt{13}$$ |
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